On the Optimal Wealth Process in a Log-Normal Market: Applications to Risk Management

34 Pages Posted: 30 Apr 2014 Last revised: 31 Jul 2015

See all articles by Phillip Monin

Phillip Monin

Board of Governors of the Federal Reserve System

Thaleia Zariphopoulou

University of Texas at Austin (Mathematics and IROM)

Date Written: June 5, 2014

Abstract

Using a stochastic representation of the optimal wealth process in the classical Merton problem, we calculate its cumulative distribution and density functions and provide bounds and monotonicity results for these quantities under general risk preferences. We also show that the optimal wealth and portfolio processes for different utility functions are related through a deterministic transformation and appropriately modified initial conditions. We analyze the value at risk (VaR) and expected shortfall of the optimal wealth process and show how each can be used to infer a CRRA investor's risk aversion coefficient. Drawing analogies to the option greeks, we study the sensitivities of the optimal wealth process with respect to the cumulative excess stock return, time, and market parameters. We conclude with a study of how sensitivities of the excess return on the optimal wealth process relate to the portfolio's beta.

Keywords: expected utility, Merton problem, value at risk, expected shortfall, portfolio Greeks

JEL Classification: G10, G11

Suggested Citation

Monin, Phillip and Zariphopoulou, Thaleia, On the Optimal Wealth Process in a Log-Normal Market: Applications to Risk Management (June 5, 2014). Available at SSRN: https://ssrn.com/abstract=2430920 or http://dx.doi.org/10.2139/ssrn.2430920

Phillip Monin (Contact Author)

Board of Governors of the Federal Reserve System ( email )

20th Street and Constitution Avenue NW
Washington, DC 20551
United States

Thaleia Zariphopoulou

University of Texas at Austin (Mathematics and IROM) ( email )

CBA 5.202
Austin, TX 78712
United States

Do you have negative results from your research you’d like to share?

Paper statistics

Downloads
119
Abstract Views
997
Rank
422,437
PlumX Metrics